Lemma 15.89.5. Let $R$ be a ring. Let $I = (f_1, \ldots , f_ n)$ be a finitely generated ideal of $R$. Let $M$ be the $R$-module generated by elements $e_1, \ldots , e_ n$ subject to the relations $f_ i e_ j - f_ j e_ i = 0$. There exists a short exact sequence

\[ 0 \to K \to M \to I \to 0 \]

such that $K$ is annihilated by $I$.

**Proof.**
This is just a truncation of the Koszul complex. The map $M \to I$ is determined by the rule $e_ i \mapsto f_ i$. If $m = \sum a_ i e_ i$ is in the kernel of $M \to I$, i.e., $\sum a_ i f_ i = 0$, then $f_ j m = \sum f_ j a_ i e_ i = (\sum f_ i a_ i) e_ j = 0$.
$\square$

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